Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have just begun to learn limits and I'm having some trouble understanding the subtraction law. I understand how to find the limit if it is something like $\lim_{x \to 2} (x+4)-(x+5)$ we just take the limits of both of the functions and then subtract them but when I do it for something like $\lim_{x \to 2} (x-4)$ I'm getting the wrong answer. What I am doing is, I am taking the limit of $x$ and then taking the limit of $-4$ and then subtracting it. The limit of $x$ is 2 and the limit of $-4$ is $-4$ but if I subtract them then I get $2-(-4) = 6$ which is incorrect. Should I be doing $2 - 4$ and negating the negative sign of $-4$?


share|cite|improve this question
When $x$ is ridiculously close to $2$ (but not equal to $2$), what is $x-4$ ridiculously close to? – André Nicolas Feb 1 '13 at 18:21
Your confusion does not come from limits, but from algebra; $x-4=x+(-4)$, not $x-(-4)$. – 1015 Feb 1 '13 at 18:38
up vote 4 down vote accepted

$\lim (x-4)=\lim x-\lim 4=2-4=-2$. In words, the limit of the difference is the difference of the limits if the latter limits exist.

Alternatively, $\lim (x-4)=\lim (x+(-4))=\lim x+\lim (-4)=2+(-4)=-2$. In words, the limit of the sum is the sum of the limits if the latter limits exist.

Your confusion arises because the negative sign has been represented twice.

share|cite|improve this answer
OR $\lim(x-4) = \lim x + \lim(-4) = 2+(-4) = -2$. – GEdgar Feb 1 '13 at 18:24
So basically, the subtraction sign between the two remains the the subtraction sign and does not effect either x or 4 but suppose it was $(x-(-4))$ then I would correct in saying that $\lim x - \lim -4$? – Jeel Shah Feb 1 '13 at 18:26

It is quite simple in this case actually. When a function $f(x)$ is continuous (like any polynomial such as $f(x)=x-4$) then by definition this means that $\lim\limits_{x \to a}f(x)=f(a)$ for every possible $a$ value. In this case that would mean $\lim\limits_{x\to 2}(x-4)=(2-4)=-2$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.